Equivalence with an Optimal Transport Problem in Two Dimensions

Abstract

In two dimensions, the least gradient problem can be interpreted in several different ways. In this Chapter, we focus on the equivalence between the least gradient problem, the Beckmann problem, and the classical Monge-Kantorovich optimal transport problem with the marginals supported on $\partial \Omega$. The main goal of this Chapter is to prove that on strictly convex domains the infimal values in the three problems coincide and there is a one-to-one correspondence between their solutions (up to an additive constant). This result is broken into several pieces to highlight that some of the equivalences hold in more general settings. We also show that there is a relationship between the respective dual problems. We continue the optimal transport approach in the next Chapter, where we present several applications of the aforementioned equivalence.